Improved Algorithms for Adaptive Compressed Sensing

In the problem of adaptive compressed sensing, one wants to estimate an approximately $k$-sparse vector $xinmathbb{R}^n$ from $m$ linear measurements $A_1 x, A_2 x,ldots, A_m x$, where $A_i$ can be chosen based on the outcomes $A_1 x,ldots, A_{i-1} x$ of previous measurements. The goal is to output a vector $hat{x}$ for which $$|x-hat{x}|_p le C cdot min_{ktext{-sparse } x} |x-x|_q,$$ with probability at least $2/3$, where $C > 0$ is an approximation factor. Indyk, Price and Woodruff (FOCS11) gave an algorithm for $p=q=2$ for $C = 1+epsilon$ with $Oh((k/epsilon) loglog (n/k))$ measurements and $Oh(log^*(k) loglog (n))$ rounds of adaptivity. We first improve their bounds, obtaining a scheme with $Oh(k cdot loglog (n/k) +(k/epsilon) cdot loglog(1/epsilon))$ measurements and $Oh(log^*(k) loglog (n))$ rounds, as well as a scheme with $Oh((k/epsilon) cdot loglog (nlog (n/k)))$ measurements and an optimal $Oh(loglog (n))$ rounds. We then provide novel adaptive compressed sensing schemes with improved bounds for $(p,p)$ for every $0 < p < 2$. We show that the improvement from $O(k log(n/k))$ measurements to $O(k log log (n/k))$ measurements in the adaptive setting can persist with a better $epsilon$-dependence for other values of $p$ and $q$. For example, when $(p,q) = (1,1)$, we obtain $O(frac{k}{sqrt{epsilon}} cdot log log n log^3 (frac{1}{epsilon}))$ measurements.